Troisième rencontre ANR HilbertXField

Notre troisième rencontre aura lieu à l’IMJ-PRG du jeudi avril 23 au vendredi avril 24, 2026. Ce sera à Jussieu, avec des conférences dans la salle 15-25 102 et des pauses café dans la salle 15-16 417. Les exposés seront donnés en anglais.

Programme provisoire:

Un dîner aura également lieu le mercredi 22 avril.

Jeudi 23 avril:

• 09h00-09h30 (15-16 417): Café et thé de bienvenue
• 09h30-10h30 (15-25 102): “From linear to metric functional analysis”, Anders Karlsson (Université de Genève)
• 10h30-11h00 (15-16 417): Pause café
• 11h00-12h00 (15-25 102): “Estimating critical exponent for Separated Anosov Representations of Free Groups”, Joaquín Lejtreger (IMJ-PRG)
• 12h00-14h00 Déjeuner à l’Ardoise, Jussieu
• 14h00-15h00 (15-25 102): “Carathéodory distance-preserving maps between bounded symmetric domains”, Bas Lemmens (Kent)
• 15h00-15h30 (15-16 417): Pause café
• 15h30-17h00 (15-25 102): Discussion
• 19h00 Diner au restaurant

Vendredi 24 avril:

• 09h30-10h30 (15-25 102): “Metric bounds for amoebas of real semialgebraic sets”, Xavier Allamigeon (Inria, Polytechnique)
• 10h30-11h00 (15-16 417): Pause café
• 11h00-12h00 (15-25 102): “Classification of representations of PGL_2(k) in the isometry groups of the infinite hyperbolic space”, Bruno Duchesne (Orsay)
• 12h00-14h00 Déjeuner à l’Ardoise, Jussieu
• 14h00-15h00 (15-25 102): “Examples of degenerations of cocompact actions in Hilbert geometry”, Balthazar Fléchelles (Université Grenoble-Alpes)
• 14h00-15h00 (15-25 102): Discussion

Liste des participants:

• Marianne Akian (Inria, Polytechnique)
• Xavier Allamigeon (Inria, Polytechnique) (orateur)
• Amanda Bigel (Inria, Polytechnique)
• Gilles Courtois (IMJ-PRG)
• Bruno Duchesne (Orsay) (orateur)
• Elisha Falbel (IMJ-PRG)
• Balthazar Fléchelles (Université Grenoble-Alpes) (orateur)
• Luca Froger (Université de Montpellier)
• Grace Garden (IMJ-PRG)
• Stéphane Gaubert (Inria, Polytechnique)
• Antonin Guilloux (IMJ-PRG)
• Anders Karlsson (Université de Genève) (orateur)
• Ricardo Katz (CONICET)
• Joaquín Lejtreger (IMJ-PRG) (orateur)
• Bas Lemmens (Kent) (orateur)
• Anne Parreau (Université Grenoble-Alpes)
• Constantin Vernicos (Université de Montpellier)
• Cormac Walsh (Inria, Polytechnique)
• Pierre Will (Université Grenoble-Alpes)

*À confirmer

Abstracts:

• Anders Karlsson (Université de Genève), “From linear to metric functional analysis”

In this talk I will describe an emerging framework — metric functional analysis — based on metric spaces and nonexpansive maps.

A central notion is metric functionals (Busemann and horofunctions), which are nonlinear analogues of linear functionals and lead to metric versions of weak topologies and compactness. There are also elements of a spectral theory: one can define analogues of operator norm and spectral radius, and obtain a spectral theorem for nonexpansive maps, as well as fixed-point and ergodic results.

These tools apply to certain nonlinear problems, but also, as it turns out, they complement the classical linear theory by removing a number of pathologies and thereby perhaps revealing structures invisible to linear techniques.  One such setting is linear operators acting on cones of positive operators equipped with invariant metrics. The discussion will naturally include works of several of the participants of the meeting.

• Joaquín Lejtreger (IMJ-PRG), “Estimating critical exponent for Separated Anosov Representations of Free Groups”

In this talk, I introduce a class of Anosov representations called strongly separated representations. I show how this condition allows one to estimate the asymptotics of the critical exponents along diverging families of such representations.

As an application, I study how the critical exponent varies along families of holonomies of convex projective structures on a pair of pants, extending an example of McMullen in rank one. This is joint work with Joaquín Lema.

• Bas Lemmens (Kent), “Carathéodory distance-preserving maps between bounded symmetric domains”

An interesting open problem in the theory of several complex variables is to find conditions under which every map between two given complex manifolds preserving the Caratheodory or Kobayashi distance is either holomorphic or anti-holomorphic. It is generally believed that as long as the domains are not biholomorphic to a Cartesian product of domains, the distance-preserving map is either holomorphic or anti-holomorphic. In this talk I will discuss some results regarding this problem in the setting of bounded symmetric domains, aka Hermitian symmetric spaces. We see will how one can exploit Jordan triple theory and the large-scale geometry of the Carathéodory distance to obtain results without posing any smoothness assumptions on the maps.

The talk is based on joint work with Cormac Walsh (Ecole Polytechnique, Paris).

• Xavier Allamigeon (Inria, Polytechnique), “Metric bounds for amoebas of real semialgebraic sets”

Denef–Pas cell decomposition provides a powerful tool for determining the valuations of semi-algebraic sets over Henselian valued fields. In this talk, I will explain how this theorem, combined with Smale’s alpha-theory, leads to bounds on the one-sided Hausdorff distance between the amoebas of a parametric family of semi-algebraic sets and its tropicalization. Joint work w/ N. Vandame.

• Bruno Duchesne (Orsay), “Classification of representations of PGL_2(k) in the isometry groups of the infinite hyperbolic space”

When k is a non-archimedean field (possibly non-local), the group PGL_2(k) has isometric actions on the infinite hyperbolic space. In a joint work with Christopher Simon, we classify such representations and prove that they form a one-parameter family. This is actually a particular case of a more general situation that also covers automorphism groups of trees of any degree (possibly infinite) and isometry groups of real hyperbolic spaces themselves.

• Balthazar Fléchelles (Université Grenoble-Alpes), “Examples of degenerations of cocompact actions in Hilbert geometry”

Degenerations of hyperbolic metrics on manifolds give rise to actions on real trees. However, Parreau, Loftin-Tamburelli-Wolf and others have shown that degenerations of convex projective structures on surfaces instead of hyperbolic structures often yield flat metrics with singularities, modeled on R^2 equipped with the hexagonal norm.

In an ongoing collaboration with Anne Parreau, we compute the explicit Gromov-Hausdorff limit of certain families of degenerations of divisible domains (with rescaled Hilbert metric) of dimension d = 2, 3 and 4 built using reflection groups à la Vinberg. The Gromov-Hausdorff limits of these degenerations are endowed with flat metrics with singularities, modeled on R^d equipped with a polytopal norm. These are the first examples in dimension > 2 of degenerations of convex projective structures that do not correspond to actions on real trees.